Abstract

Two-beam interferogram intensity modulation decoding using spatial carrier phase shifting interferometry is discussed. Single frame recording, simplicity of experimental equipment, and uncomplicated data processing are the main advantages of the method. A comprehensive analysis of the influence of systematic errors (spatial carrier miscalibration, nonuniform average intensity profile, and nonlinear recording) on the modulation distribution determination using automatic fringe pattern analysis techniques is presented. The results of searching for the optimum calculation algorithm are described. Extensive numerical simulations are compared with laboratory findings obtained when testing vibrating silicon microelements under various experimental conditions.

© 2007 Optical Society of America

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References

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  2. K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt. Eng. 44, 065601 (2005).
    [CrossRef]
  3. K. Patorski and A. Styk, "Interferogram intensity modulation calculations using temporal phase shifting: error analysis," Opt. Eng. 45, 085602 (2006).
    [CrossRef]
  4. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Dekker, 1998).
  5. S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, "3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope," Opt. Lasers Eng. 36, 77-101 (2001).
    [CrossRef]
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    [CrossRef]
  7. S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, "Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization," Proc. SPIE 4400, 51-60 (2001).
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2006 (1)

K. Patorski and A. Styk, "Interferogram intensity modulation calculations using temporal phase shifting: error analysis," Opt. Eng. 45, 085602 (2006).
[CrossRef]

2005 (1)

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt. Eng. 44, 065601 (2005).
[CrossRef]

2001 (2)

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, "3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope," Opt. Lasers Eng. 36, 77-101 (2001).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, "Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization," Proc. SPIE 4400, 51-60 (2001).
[CrossRef]

1997 (1)

1996 (2)

K. G. Larkin, "Efficient nonlinear algorithm for envelope detection in white light interferometry," J. Opt. Soc. Am. A 13, 832-843 (1996).
[CrossRef]

K. Creath and J. Schmitt, "N-point spatial phase-measurement techniques for non-destructive testing," Opt. Lasers Eng. 24, 365-379 (1996).
[CrossRef]

1995 (2)

1987 (1)

1983 (1)

Appl. Opt. (4)

J. Mod. Opt. (1)

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

K. Patorski, Z. Sienicki, and A. Styk, "The phase shifting method contrast calculations in time average interferometry: error analysis," Opt. Eng. 44, 065601 (2005).
[CrossRef]

K. Patorski and A. Styk, "Interferogram intensity modulation calculations using temporal phase shifting: error analysis," Opt. Eng. 45, 085602 (2006).
[CrossRef]

Opt. Lasers Eng. (2)

S. Petitgrand, R. Yahiaoui, K. Danaie, A. Bosseboeuf, and J. P. Gilles, "3D measurement of micromechanical devices vibration mode shapes with a stroboscopic interferometric microscope," Opt. Lasers Eng. 36, 77-101 (2001).
[CrossRef]

K. Creath and J. Schmitt, "N-point spatial phase-measurement techniques for non-destructive testing," Opt. Lasers Eng. 24, 365-379 (1996).
[CrossRef]

Proc. SPIE (1)

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, "Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization," Proc. SPIE 4400, 51-60 (2001).
[CrossRef]

Other (2)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Dekker, 1998).

D. W. Robinson and G. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics, 1993).

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Figures (17)

Fig. 1
Fig. 1

Simulated exemplary fringe patterns with Gaussian changes of the intensity modulation distribution (a) perpendicular and (b) parallel mutual orientation with respect to the interferogram carrier fringes.

Fig. 2
Fig. 2

(Color online) Simulated interferogram contrast envelope functions across the interferograms.

Fig. 3
Fig. 3

(Color online) Differences between the error-free and calculated modulations using algorithms 3 A mod , 5 A mod , and 5L mod . The contrast envelope was described by linear function, first row; Gaussian function, second row; and Bessel function J 0 , third row. The case of perpendicular orientation between the direction of interferogram carrier fringes and modulation changes.

Fig. 4
Fig. 4

(Color online) Difference between the error-free modulation and modulation maps calculated using algorithms 3 A mod , 5 A mod , and 5 L mod . The contrast envelope was described by a Gaussian function. The case of parallel mutual orientation between the interferogram carrier fringes and the modulation change direction.

Fig. 5
Fig. 5

(Color online) Difference between the error-free modulation distribution and the modulation calculated using algorithms 3 A mod , 5 A mod , and 5L mod . The interferogram bias distribution described by a Gaussian function; perpendicular orientation between the interferogram carrier fringes and interferogram bias change direction.

Fig. 6
Fig. 6

(Color online) Difference between the error-free modulation and modulation distributions calculated using three algorithms considered with the presence of the carrier frequency miscalibration. Phase shift between consecutive pixels is equal to 60°, first row, and 120°, second row. Uniform modulation distribution was assumed.

Fig. 7
Fig. 7

(Color online) Cross sections through the difference between the error-free modulation and the evaluated modulation distributions calculated using three algorithms under consideration. Phase shift between consecutive pixels: 60°, first row; 120°, second row. Modulation distribution changes described by the Bessel function J 0 . The case of perpendicular mutual orientation between interferogram carrier fringes and the interferogram modulation change direction.

Fig. 8
Fig. 8

(Color online) Calibration curves showing change of modulation values evaluated when different spatial carrier frequency is applied (spatial frequency is rescaled to the relative phase shift between adjacent pixels).

Fig. 9
Fig. 9

(Color online) Cross sections through the difference between the error-free modulation and modulation distributions calculated using algorithms 3 A mod , 5 A mod , and 5L mod showing the influence of a nonlinear carrier frequency miscalibration. See text for details.

Fig. 10
Fig. 10

(Color online) Error distribution in the interferogram intensity modulation evaluated using algorithms 3 A mod , 5 A mod , and 5L mod in the presence of a nonlinear recording error. Modulation changes according to Bessel function J 0 ; the case of interferogram carrier fringes running perpendicularly to the interferogram modulation change direction.

Fig. 11
Fig. 11

(Color online) Horizontal cross sections through the modulation maps calculated using considered algorithms in the presence of nonlinear recording. Modulation changes according to the Bessel function J 0 . The case of parallel mutual orientation between interferogram carrier fringes and the interferogram modulation change direction.

Fig. 12
Fig. 12

(Color online) Error in the modulation distribution calculated using three considered algorithms with the presence of a nonlinear recording error and carrier frequency miscalibration. Constant modulation distribution has been assumed.

Fig. 13
Fig. 13

Two-beam interferogram intensity modulation distributions for the static state of the AFM cantilevers calculated using three considered algorithms. Applied carrier frequency was set to the value of four pixels per fringe.

Fig. 14
Fig. 14

First row shows two-beam interferogram intensity modulation distributions for the nonvibrating AFM cantilevers calculated using three considered algorithms. Applied carrier frequency was approximately six pixels per fringe. The second row shows the magnified upper left corners of the calculated modulation maps.

Fig. 15
Fig. 15

Two-beam interferogram intensity modulation distributions for the first resonance mode of an AFM cantilever vibrating at 21.1   kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.

Fig. 16
Fig. 16

Two-beam interferogram intensity modulation distributions for the second resonance mode of an AFM cantilever vibrating at 142   kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.

Fig. 17
Fig. 17

Two-beam interferogram intensity modulation distributions for the first torsional mode of an AFM cantilever vibrating at 175   kHz calculated by three investigated algorithms; applied carrier frequency was set to four pixels per fringe.

Tables (2)

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Table 1 Carrier Fringes Perpendicular to the Direction of Modulation Changes

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Table 2 Carrier Fringes Parallel to the Direction of Modulation Changes

Equations (6)

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I ( x , y ) = I 0 ( x , y ) { 1 + K ( x , y ) γ ( x , y ) cos ( φ ( x , y ) + α ( x , y ) ) } ,
3 A mod : 1 2 ( I 2 I 4 ) 2 + 4 I 3 2 ( I 2 + I 4 ) 2 ,
I i = I i 0.5 ( I i 1 + I i + 1 ) , i = 2 , 3 , 4 ;
5 A mod : 1 4 4 ( I 2 I 4 ) 2 + ( 2 I 3 I 1 I 5 ) 2 ;
5 L mod : 1 2 ( I 2 I 4 ) 2 ( I 1 I 3 ) ( I 3 I 5 ) ,
I = I + a I 2 + b I 3 + c I 4 + ,

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